Character naming in Synecdoche New York

I think what you're saying is right. Caden is so obsessed with the inevitability of death that he sees it everywhere. He complains of being unwell since the start of the film but actually outlives most of the other characters. He sees others decaying as well, most notably in the progressively worse skin lesions on his psychiatrist's foot.

Another occurrence of Cotard's syndrome is when Adele tells young Olive that she doesn't have blood. Caden vocally disagrees with this, presumably not wanting his child to deny her mortality.

I've found another link to the real Cotard through the IMDB FAQ, which states:

Hazel's books also have symbolic resonance - Swann's Way is the first volume of In Search of Lost Time, an epic novel by Marcel Proust in which the author explores and interprets the events of his own life. A character in it is named Dr Cottard, who is based on Dr Jules Cotard, a French neurologist who described the Cotard delusion, a patient's delusional belief that they are dead, do not exist or do not have bodily organs. The main character of Synecdoche, New York is also named Cotard, and perhaps he believes that humans have the delusional belief that they are alive and that they exist!